A lump of bomb steel and a sacred branch are required to make a new katana. You will have to ask Jaucribaix's pupils to find out where to get these items.

Walkthrough

Speak with Jaucribaix in Norg. He wants a branch from a tree in a sacred place and iron filings from a bomb. Talk to Ranemaud to receive a sacred tree sapling. Now, talk to Aeka to receive old steel from the east. The next two parts can be done in any order. Trade the steel to the ??? at the far west of Konschtat Highlands at D-8. A bomb enemy, Forger, will appear. Kill it and you will receive the Bomb Steel. However, know that if you fight it with a party, Forger only drops 1 item at a time, and you can only trade the iron to the ??? every 3 minutes. For the other part, use a hatchet at the ??? point on the huge tree at the east side of the Sanctuary of Zi'Tah at L-10. A walking tree enemy, Guardian Treant, will appear. Kill it and trade your sapling to the ??? again to receive the branch. Return to Jaucribaix with these items. After you give them to him, he will tell you to come back in three days (3 real-life hours). When you return, you will receive the Samurai Advanced Job and an Unsigned Sword.

It is ABSOLUTELY IMPERATIVE that you do not lose the Mumeito you are given unless you never, ever plan on getting the Samurai Artifact Armor. Having to replace the sword can cost you up to 100k gil.